Lesson 79 — Deep Dive into Bayes' Theorem

$P(A) = 0.3$, $P(B) = 0.5$, $P(A \cap B) = 0.15$. Calculate $P(A \mid B)$.

$P(A \mid B) = 0.6$, $P(B) = 0.5$. Calculate $P(A \cap B)$.

$P(A) = 0.1$, $P(B \mid A) = 0.8$, $P(B \mid \bar A) = 0.2$. Calculate $P(B)$.

With the data from exercise 79.3, calculate $P(A \mid B)$.

Disease with 0.5% prevalence. Diagnostic test: 95% sensitivity, 95% specificity. Calculate the PPV using frequencies in 10,000 people.

Same data as exercise 79.5, but with 50% prevalence. Calculate the PPV and compare with the previous result.

Spam filter: $P(\text{spam}) = 0.3$. Word "FREE" appears in 60% of spams and 5% of legitimate emails. Calculate $P(\text{spam} \mid \text{FREE})$.

Urn A: 2 red, 3 blue. Urn B: 5 red, 1 blue. An urn is chosen at random and a red ball is drawn. What is the probability the urn is A?

3 coins: 2 fair, 1 double-headed. One is chosen at random, flipped once, comes up heads. What is the probability the chosen coin is the double-headed one?

$P(\text{smoker}) = 0.2$. $P(\text{cancer} \mid \text{smoker}) = 0.1$. $P(\text{cancer} \mid \neg\text{smoker}) = 0.01$. Given a person has cancer, what is the probability they are a smoker?

Sequential updating: two positive tests with 90% sensitivity and 90% specificity, applied to a disease with 1% prevalence. Use the posterior of the 1st test as the prior of the 2nd. What is the PPV after both consecutive positive results?

For a test with 90% sensitivity and 95% specificity, calculate the positive likelihood ratio $\text{LR}^+ = \text{sens}/(1 - \text{spec})$.

Prior odds of 1:99 (1% prevalence). $\text{LR}^+ = 18$ (90% sensitivity, 95% specificity). Calculate the posterior odds and the posterior.

Which of the following values is the correct posterior in a context with prior odds 1:99 and $\text{LR}^+ = 18$?

Prior $\theta \sim \text{Beta}(2, 2)$. 7 heads observed in 10 flips. Determine the posterior.

Prior $\theta \sim \text{Beta}(1, 1)$ (uniform). 0 heads observed in 5 flips. Determine the posterior and its mean.

In exercise 79.15, what is the posterior mean?

Prior $\theta \sim \text{Beta}(2, 8)$. New batch: 30 parts inspected, 6 defective. Determine the posterior and posterior mean.

COVID-19 in endemic phase: 5% prevalence. Rapid test: 80% sensitivity, 95% specificity. Calculate the PPV using frequencies in 10,000 people. Is it worth automatically isolating all positives?

Naive Bayes for email: $P(\text{spam}) = 0.3$. In training: "FREE" appears in 60% of spams and 5% of hams; "won" appears in 50% of spams and 10% of hams. An email contains both words. Classify assuming conditional independence.

Three diseases: A (10% in population), B (5%), C (1%). Patient presents symptom S with $P(S|A) = 0.3$, $P(S|B) = 0.9$, $P(S|C) = 0.9$. Which disease is most likely?

Prosecutor's fallacy: DNA evidence has a frequency of 1/1000 in the population. The prosecutor claims the probability of innocence is 1/1000. Why is this reasoning wrong? Calculate the correct posterior assuming there are 100,000 plausible suspects in the city.

Fraud classifier: 95% sensitivity, 99.9% specificity. Frauds: 0.1% of transactions. Calculate the PPV. How many false positives for every true positive?

Pregnancy test: 99% sensitivity, 98% specificity. Woman with prior probability of pregnancy of 30%. Calculate the PPV.

Polygraph: 70% sensitivity, 80% specificity. In interrogation with a suspect who has a 5% prior of guilt. Calculate the posterior after a positive result. Is the result admissible as sufficient evidence to convict?

Two independent positive tests (sens$_1$ = 0.9, spec$_1$ = 0.95; sens$_2$ = 0.85, spec$_2$ = 0.90). Prevalence 2%. Calculate the posterior after both positive results via sequential updating.

In a lineup, one suspect has red hair (H) with a 70% probability of being the culprit. A witness identifies the red-haired one with 90% probability when the culprit is H, and erroneously 15% of the time when the culprit is not H. Given the witness pointed to H, what is the posterior of guilt?

Quality control with 3 lines (A: 40% of production, 2% defect; B: 35%, 3%; C: 25%, 5%). A defective part is found. Determine the probability of each line being the origin.

What is the base rate fallacy?

Why does the prior matter even in "objective science"? An analysis that ignores the prior is equivalent to what implicit assumption?

Two independent positive tests with likelihood ratios $r_1$ and $r_2$. What is the effect on the odds form?

What is the practical difference between using a Beta(1,1) prior and a Beta(10,10) prior for a coin? In which case will the posterior be more sensitive to new data?

Show that two conditionally independent positive tests given $H$ result in posterior odds equal to $r_1 \times r_2 \times$ prior odds, where $r_i = \text{LR}_i^+$.

Demonstrate that the posterior of the Bernoulli-Beta model is Beta($\alpha + k$, $\beta + n - k$) when the prior is Beta($\alpha$, $\beta$) and we observe $k$ successes in $n$ trials.

Demonstrate Bayes' theorem from the definition of conditional probability and the law of total probability.

Show that $P(A \mid B) = P(B \mid A)\,P(A)/P(B)$ using only the definition of conditional probability. Identify why $P(A \mid B) \neq P(B \mid A)$ in general.

Monty Hall problem with 3 doors. Use Bayes to calculate the probability of the car being in each door after Monty (who knows where the car is) opens an empty door. Should you switch?

In Naive Bayes with binary features, show that the classifier is equivalent to multiplying the individual LRs of each feature. What happens when the conditional independence assumption is violated?

Demonstrate that the odds form of Bayes, posterior odds = LR $\times$ prior odds, follows directly from the usual form of Bayes' theorem for two complementary events $H$ and $\neg H$.

Show that the mean of the posterior Beta($\alpha + k$, $\beta + n - k$) converges to the maximum likelihood estimator $k/n$ when $n \to \infty$, for any fixed prior Beta($\alpha$, $\beta$). What does this imply about the relationship between Bayes and frequentism for large samples?